This work reports on results from two Recoil-Distance-Doppler-Shift lifetime measurements of excited states in 155Dy and 178Os. The experiments were carried out at the GASP spektrometer of the Laboratori Nazional i di Legnaro in combination with the Cologne plunger apparatus. The main purpose of the performed experiments was to test the predictions of the X(5) critical point symmetry in these two nuclei. In 156Dy and 178Os 29 lifetimes of excited states were derived using the Differe ntial-Decay-Curve method. In weaker reaction channels the nuclei 155Dy, 157Dy and 177Os were populated. In these nuclei 32 additional lifetimes were measured, most of them for the first time. In order to calculate absolute transition probabilities from the measured lifetimes of the first excited band in 156Dy, essential branching ratios were derived from the measured data with a very small systematic error (<5%). The most important results can be summarized as mentioned below: Constant quadrupole moments in the ground band of 156Dy do not support the results from [EML84] and remove a big discrepancy in the discussion of shape transitions in this nucleus. Additionally these results point out that 156Dy can not be described at the same quality by the model of the X(5) symmetry as neighboring N=90 isotones. Nevertheless, lifetimes measured in the first excited band, confirm that this nucleus can be located close to the critical point X(5). With model calculations, special criteria of the X(5) model were found that can be used to identify other X(5)-like nuclei. Using these criterias a new region of X(5)-like nuclei could be suggested within the osmium isotopes in the A=180 mass region. The measured lifetimes in 178Os confirm the consistency of a X(5) description in these nuclei. A comparision with the well established X(5)-like nuclei in the N=90 isotones gives an agreement with the X(5) description of at least the same quality.